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Wave propagation in a network of extended Morris–Lecar neurons with electromagnetic induction and its local kinetics

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Abstract

An extended Morris–Lecar neuron model incorporating electromagnetic flux coupling and external excitation is proposed. The complete dynamical behavior of the new model is investigated as autonomous and non-autonomous forms. Multistability and coexisting attractors are shown for the new neuron model. It is shown that when external excitation is introduced, the complete dynamical behavior of the system changes compared to the non-excited version. To study the wave propagation pattern, we construct a network of the extended neuron model and study the generation of spiral waves under various conditions of coupling strength, stimuli parameters and system parameters. We have also studied the wave propagation pattern considering the time-based chaotic-periodic neurons in the network.

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References

  1. FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 17, 257–278 (1955)

    Google Scholar 

  2. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)

    Google Scholar 

  3. Fitz, H.R.: Chapter 1: Mathematical models of excitation and propagation in nerve. In: Schwan, H.P. (ed.) Biological Engineering, pp. 1–85. McGraw-Hill Book Co., Nork York (1969)

    Google Scholar 

  4. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc IRE. 50, 2061–2070 (1962)

    Google Scholar 

  5. Hodgkin, A., Huxley, A.: Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J. Physiol. 116(4), 449–472 (1952)

    Google Scholar 

  6. Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35(1), 193–213 (1981)

    Google Scholar 

  7. Chen, P.-L.: Some aspects of the Morris–Lecar model and the myelinated axon models with Morris–Lecar dynamics. Math. Comput. Modell. 17(8), 85–97 (1993)

    MathSciNet  MATH  Google Scholar 

  8. Tsumoto, K., Kitajima, H., Yoshinaga, T., Aihara, K., Kawakami, H.: Bifurcations in Morris–Lecar neuron model. Neurocomputing 69, 293–316 (2006)

    Google Scholar 

  9. Wang, H.X., Lu, Q.S., Wang, Q.Y.: Generation of firing rhythm patterns and synchronization in the Morris–Lecar neuron model. Int. J. Nonlinear Sci. Numer. Simul. 6, 7–12 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Wang, H.X., Lu, Q.S., Wang, Q.Y.: Bursting and synchronization transition in the coupled modified M–L neurons. Commun. Nonlinear Sci. Numer. Simul. 13, 1668–1675 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Xinlin, S., Hengtong, W., Yong, C.: Autapse-induced firing patterns transitions in the Morris–Lecar neuron model. Nonlinear Dyn. (2019). https://doi.org/10.1007/s11071-019-04925-7

    Article  Google Scholar 

  12. Farjami, S., Kirk, V., Osinga, H.M.: Interactions between a locally separating stable manifold and a bursting periodic orbit. Eur. Phys. J. Spec. Top. 227, 603–614 (2018)

    Google Scholar 

  13. Vetriveeran, R., Maheshwar, P.D.S., Zubaer, I.M., Hyongsuk, K., Leon, C.: Third-order memristive Morris–Lecar model of barnacle muscle fiber. Int. J. Bifurc. Chaos 27(4), 1730015 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Congmin, L., Xuanliang, L., Shenquan, L.: Bifurcation analysis of a Morris–Lecar neuron model. Biol. Cybern. 108, 75–84 (2014)

    MathSciNet  Google Scholar 

  15. Prescott, S.A., De Koninck, Y., Sejnowski, T.J.: Biophysical basis for three distinct dynamical mechanisms of action potential initiation. PLOS Comput. Biol. 4, 1–18 (2008)

    MathSciNet  Google Scholar 

  16. Bocheng, B., Qinfeng, Y., Lei, Z., Han, B., Quan, X., Yajuan, Y., Mo, C.: Chaotic bursting dynamics and coexisting multistable firing patterns in 3D autonomous Morris–Lecar model and microcontroller-based validations. Int. J. Bifurc. Chaos 29(10), 1950134–1950152 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Alexandre, W., Miguel, A.F.S., Jose, M.C., Kazuyuki, A.: Building electronic bursters with the Morris–Lecar neuron model. Int. J. Bifurc. Chaos 16(12), 3617–3630 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Jiang, W., Meili, L., Huiyan, L.: Synchronization of coupled equations of Morris–Lecar model. Commun. Nonlinear Sci. Numer. Simul. 13, 1169–1179 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst. 5(4), 380–385 (2002)

    Google Scholar 

  20. Shima, S., Kuramoto, Y.: Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. Phys. Rev. E 69, 036213 (2004)

    Google Scholar 

  21. Kuramoto, Y., Shima, S.: Rotating spirals without phase singularity in reaction-diffusion systems. Prog. Theor. Phys. Suppl. 150, 115 (2003)

    MathSciNet  Google Scholar 

  22. Kuramoto, Y., Shima, S., Battogtokh, D., Shiogai, Y.: Mean-field theory revives in self-oscillatory fields with non-local coupling. Prog. Theor. Phys. Suppl. 161, 127 (2006)

    Google Scholar 

  23. Erik, A.M., Carlo, R.L., Steven, H.S.: Solvable model of spiral wave chimeras. Phys. Rev. Lett. 3, 150 (2010)

    Google Scholar 

  24. Samie, F.H., Mandapati, R., Gray, R.A., Watanabe, Y., Zuur, C.: A mechanism of transition from ventricular fibrillation to tachycardia. Circ. Res. 86, 684–691 (2000)

    Google Scholar 

  25. Samie, F.H., Jalife, J.: Mechanisms underlying ventricular tachycardia and its transition to ventricular fibrillation in the structurally normal heart. Cardiovasc. Res. 50, 242–250 (2001)

    Google Scholar 

  26. Yuan, G.Y., Wang, G.R., Chen, S.G.: Control of spiral waves and spatiotemporal chaos by periodic perturbation near the boundary. Europhys. Lett. 72, 908 (2005)

    Google Scholar 

  27. Fenton, F.H., Cherry, E.M., Hastings, H.M., Evans, S.J.: Multiple mechanisms of spiral wave breakup. Chaos 12, 852–892 (2002)

    Google Scholar 

  28. Aranson, I., Kessler, D., Mitkov, I.: Drift of spiral wave in excitable media. Physica D 85, 142–155 (1995)

    MATH  Google Scholar 

  29. Zykov, V.S.: Kinematics of rigidly rotating spiral waves. Physica D 238, 931–940 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Heather, A., Brooks, P.C.B.: Quasicycles in the stochastic hybrid Morris–Lecar neural model. Phys. Rev. E 92, 012704 (2015)

    MathSciNet  Google Scholar 

  31. Hou, Z.H., Xin, H.W.: Noise-sustained spiral waves: effect of spatial and temporal memory. Phys. Rev. Lett. 89, 280601 (2002)

    Google Scholar 

  32. Gu, H.G., Jia, B., Li, Y.Y., Chen, G.R.: White noise-induced spiral waves and multiple spatial coherence resonances in a neuronal network with type I excitability. Physica A 67, 169 (2013)

    MathSciNet  Google Scholar 

  33. Ali, C., Philipp, H., Mahmut, O., Muhammet, U.: Chimera states in networks of type-I Morris–Lecar neurons. Phys. Rev. E 98, 062217 (2018)

    MathSciNet  Google Scholar 

  34. Xinyi, W., Jun, M.: The formation mechanism of defects, spiral wave in the network of neurons. Plos One 8, 1 (2013)

    Google Scholar 

  35. Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 10, 1171–1266 (2000)

    MathSciNet  MATH  Google Scholar 

  36. Lv, M., Ma, J.: Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomput 205(3), 75–81 (2016)

    Google Scholar 

  37. Karthikeyan, R., Fahimeh, N., Anitha, K., Ahmed, A., Tasawar, H., Viet-Thanh, P.: Dynamics of a neuron exposed to integer order and fractional order discontinuous external magnetic flux. Front. Inf. Technol. Electron. Eng. 20, 584–590 (2019)

    Google Scholar 

  38. Fatemeh, P., Karthikeyan, R., Fawaz, E.A., Tasawar, H., Pham, V.-T., Iqtadar, H.: Birth and Death of spiral waves in a network of Hindmarsh-Rose neurons with exponential magnetic flux and excitable media. Appl. Math. Comput. 354, 377–384 (2019)

    MathSciNet  MATH  Google Scholar 

  39. Karthikeyan, R., Fatemeh, P., Hamed, A., Boshra, H., Sajad, J., Vesna, B.: Spiral waves in externally excited neuronal network: solvable model with a monotonically differentiable magnetic flux. Chaos Interdiscip. J. Nonlinear Sci. 29(4), 043109 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Rajagopal, K., Khalaf, A.J.M., Parastesh, F., et al.: Dynamical behavior and network analysis of an extended Hindmarsh-Rose neuron model. Nonlinear Dyn 98, 477–487 (2019)

    MATH  Google Scholar 

  41. Jun, M., Ya, W., Chunni, W., Ying, X., Guodong, R.: Mode selection in electrical activities of myocardial cell exposed to electromagnetic radiation. Chaos Solitons Fractals 99, 219–225 (2017)

    Google Scholar 

  42. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D Nonlinear Phenom. 16, 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  43. Wu, F., Ma, J., Zhang, G.: Energy estimation and coupling synchronization between biophysical neurons. Sci. China Technol. Sci. 63, 625–636 (2020)

    Google Scholar 

  44. Ma, J., Yang, Z., Yang, L., Tang, J.: A physical view of computational neurodynamics. J. Zhejiang Univ. Sci. A 20(9), 639–659 (2019)

    Google Scholar 

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Authors and Affiliations

  1. Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Karthikeyan Rajagopal

  2. Mathematical Institute, University of Oxford, Andrew Wiles Building, Oxford, UK

    Irene Moroz

  3. Center for Nonlinear Dynamics, Defence University, Bishoftu, Ethiopia

    Anitha Karthikeyan

  4. Center for Nonlinear Dynamics and Control, Mekelle University, Mekelle, Ethiopia

    Prakash Duraisamy

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  1. Karthikeyan Rajagopal
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  2. Irene Moroz
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  3. Anitha Karthikeyan
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  4. Prakash Duraisamy
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Correspondence to Karthikeyan Rajagopal.

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Rajagopal, K., Moroz, I., Karthikeyan, A. et al. Wave propagation in a network of extended Morris–Lecar neurons with electromagnetic induction and its local kinetics. Nonlinear Dyn 100, 3625–3644 (2020). https://doi.org/10.1007/s11071-020-05643-1

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